Application and Practice Questions

Lesson 2 has thus far focused on how to analyze motion situations using the work and energy relationship. The relationship could be summarized by the following statements:

There is a relationship between work and mechanical energy change. Whenever work is done upon an object by an external or nonconservative force, there will be a change in the total mechanical energy of the object. If only internal forces are doing work (no work done by external forces), there is no change in total mechanical energy; the total mechanical energy is said to be "conserved." The quantitative relationship between work and the two forms of mechanical energy is expressed by the following equation:

KEi + PEi + Wext = KEf + PEf

Now an effort will be made to apply this relationship to a variety of motion scenarios in order to test our understanding.

Check Your Understanding

Use your understanding of the work-energy theorem to answer the following questions. Then click the button to view the answers.

1. Consider the falling and rolling motion of the ball in the following two resistance-free situations. In one situation, the ball falls off the top of the platform to the floor. In the other situation, the ball rolls from the top of the platform along the staircase-like pathway to the floor. For each situation, indicate what types of forces are doing work upon the ball. Indicate whether the energy of the ball is conserved and explain why. Finally, fill in the blanks for the 2-kg ball.

The only force doing work is gravity. Since it is an internal or conservative force, the total mechanical energy is conserved. Thus, the 100 J of original mechanical energy is present at each position. So the KE for A is 50 J.

The PE at the same stairstep is 50 J (C) and thus the KE is also 50 J (D).

The PE at zero height is 0 J (F and I). And so the kinetic energy at the bottom of the hill is 100 J (G and J).

Using the equation KE = 0.5*m*v2, the velocity can be determined to be 7.07 m/s for B and E and 10 m/s for H and K.

The answers given here for the speed values are presuming that all the kinetic energy of the ball is in the form of translational kinetic energy. In actuality, some of the kinetic energy would be in the form of rotational kinetic energy. Thus, the actual speed values would be slightly less than those indicated. (Rotational kinetic energy is not discussed here at The Physics Classroom Tutorial.)

2. If frictional forces and air resistance were acting upon the falling ball in #1 would the kinetic energy of the ball just prior to striking the ground be more, less, or equal to the value predicted in #1?

The answer is D. The total mechanical energy (i.e., the sum of the kinetic and potential energies) is everywhere the same whenever there are no external or nonconservative forces (such as friction or air resistance) doing work.

4. The object will have a minimum gravitational potential energy at point ____.

The answer is C. Since the total mechanical energy is conserved, kinetic energy (and thus, speed) will be greatest when the potential energy is smallest. Point B is the only point that is lower than point C. The reasoning would follow that point B is the point with the smallest PE, the greatest KE, and the greatest speed. Therefore, the object will have less kinetic energy at point C than at point B (only).

6. Many drivers' education books provide tables that relate a car's braking distance to the speed of the car (see table below). Utilize what you have learned about the stopping distance-velocity relationship to complete the table.

The car skids 135 m. There is a three-fold increase in the speed of the car (150 / 50 = 3). Thus, there must be a nine-fold increase in the stopping distance. Multiply 15 meters by 9.

8. Two baseballs are fired into a pile of hay. If one has twice the speed of the other, how much farther does the faster baseball penetrate? (Assume that the force of the haystack on the baseballs is constant).

The faster baseball penetrates four times as far. When there is a two-fold increase in speed, there is a four-fold increase in stopping distance. For constant resistance forces, stopping distance is proportional to the square of the speed.

9. Use the law of conservation of energy (assume no friction) to fill in the blanks at the various marked positions for a 1000-kg roller coaster car.

10. If the angle of the initial drop in the roller coaster diagram above were 60 degrees (and all other factors were kept constant), would the speed at the bottom of the hill be any different? Explain.

The angle does not affect the speed at the bottom of the incline. The speed at the bottom of the incline is dependent upon the initial height of the incline. Many students believe that a smaller angle means a smaller speed at the bottom. But such students are confusing speed with acceleration. A smaller angle will lead to a smaller acceleration along the incline.

The total amount of mechanical energy is conserved in free-fall situations (no external forces doing work). Thus, the potential energy that is lost is transformed into kinetic energy. The object loses 200 J of potential energy (PE loss = m * g * h where the m•g is 200 N (i.e., the object's weight).

Observe that a confusion of mass (~20 kg) and weight (200 N) will inevitably lead to the wrong answer.

14. A rope is attached to a 50.0-kg crate to pull it up a frictionless incline at constant speed to a height of 3-meters. A diagram of the situation and a free-body diagram are shown below. Note that the force of gravity has two components (parallel and perpendicular component); the parallel component balances the applied force and the perpendicular component balances the normal force.

(KEi = KEf since speed is constant. Thus, both KE terms can be eliminated from the equation.)

Wext = (50 kg) * (9.8 m/s/s) * (3 m) = 1470 J

We Would Like to Suggest ...

Sometimes it isn't enough to just read about it. You have to interact with it! And that's exactly what you do when you use one of The Physics Classroom's Interactives. We would like to suggest that you combine the reading of this page with the use of our Roller Coaster Model Interactive, our Mass on a Spring Interactive, and/or our Chart That Motion Interactive. These three Interactives can be found in the Physics Interactive section of our website and provide an interactive opportunity to explore the work-energy relationship.